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Eigenvalue bounds for the distance-$t$ chromatic number of a graph and their application to Lee codes

Aida Abiad, Alessandro Neri, Luuk Reijnders

TL;DR

The paper develops eigenvalue-based Ratio-type bounds for the distance-$t$ chromatic number $\chi_t(G)$ and extends them to non-regular graphs, yielding optimized bounds and applications to coding theory. It provides alternative spectral proofs for hypercube bounds and derives new bounds for $\chi_t(Q_n)$ with $t=4,5$, while also analyzing Lee graphs $G(n,q)$ to bound $\chi_2(G(n,q))$ using spectral and number-theoretic tools. A novel connection between the Lee metric and graph spectra enables a complete characterization of when perfect Lee codes with minimum distance $3$ exist, tying code existence to divisibility conditions and to the spectrum of Lee graphs. Overall, the work demonstrates that spectral methods effectively capture the structure of the Lee metric and yield sharp, interpretable results for both bounds and nonexistence assertions in Lee codes.

Abstract

We derive eigenvalue bounds for the $t$-distance chromatic number of a graph, which is a generalization of the classical chromatic number. We apply such bounds to hypercube graphs, providing alternative spectral proofs for results by Ngo, Du and Graham [Inf. Process. Lett., 2002], and improving their bound for several instances. We also apply the eigenvalue bounds to Lee graphs, extending results by Kim and Kim [Discrete Appl. Math., 2011]. Finally, we provide a complete characterization for the existence of perfect Lee codes of minimum distance $3$. In order to prove our results, we use a mix of spectral and number theory tools. Our results, which provide the first application of spectral methods to Lee codes, illustrate that such methods succeed to capture the nature of the Lee metric.

Eigenvalue bounds for the distance-$t$ chromatic number of a graph and their application to Lee codes

TL;DR

The paper develops eigenvalue-based Ratio-type bounds for the distance- chromatic number and extends them to non-regular graphs, yielding optimized bounds and applications to coding theory. It provides alternative spectral proofs for hypercube bounds and derives new bounds for with , while also analyzing Lee graphs to bound using spectral and number-theoretic tools. A novel connection between the Lee metric and graph spectra enables a complete characterization of when perfect Lee codes with minimum distance exist, tying code existence to divisibility conditions and to the spectrum of Lee graphs. Overall, the work demonstrates that spectral methods effectively capture the structure of the Lee metric and yield sharp, interpretable results for both bounds and nonexistence assertions in Lee codes.

Abstract

We derive eigenvalue bounds for the -distance chromatic number of a graph, which is a generalization of the classical chromatic number. We apply such bounds to hypercube graphs, providing alternative spectral proofs for results by Ngo, Du and Graham [Inf. Process. Lett., 2002], and improving their bound for several instances. We also apply the eigenvalue bounds to Lee graphs, extending results by Kim and Kim [Discrete Appl. Math., 2011]. Finally, we provide a complete characterization for the existence of perfect Lee codes of minimum distance . In order to prove our results, we use a mix of spectral and number theory tools. Our results, which provide the first application of spectral methods to Lee codes, illustrate that such methods succeed to capture the nature of the Lee metric.
Paper Structure (11 sections, 36 theorems, 59 equations, 1 figure, 2 tables)

This paper contains 11 sections, 36 theorems, 59 equations, 1 figure, 2 tables.

Table of Contents

  1. Keywords:
  2. Introduction
  3. Eigenvalue bounds for the distance chromatic number of a graph
  4. Optimization of the Ratio-type bounds
  5. Optimality of the Ratio-type bounds for $t=2,3$
  6. Bounding the distance chromatic number of the hypercube graph
  7. The Lee metric and the Lee graph
  8. Combinatorial properties of the Lee graph
  9. Spectral properties of the Lee graph
  10. Bounding the distance chromatic number of the Lee graph
  11. A characterization of perfect Lee codes with minimum distance $3$

Key Result

Theorem 1

Let $G$ be a graph with $n$ vertices and eigenvalues $\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_n$ and adjacency matrix $A$. Let $p \in \mathbb{R}_t[x]$ with corresponding parameters $W(p) := \max_{u \in V}$$\{(p(A))_{uu}\}$ and $\lambda(p):= \min_{i\in[2,n]} \{p(\lambda_i)\}$, and assume $p

Figures (1)

  • Figure 1: The Lee graph $G(2,4)$.

Theorems & Definitions (56)

  • Theorem 1: General Ratio-type bound ACFNS2022
  • Theorem 2: Ratio-type bound
  • proof
  • Theorem 3
  • proof
  • Corollary 4
  • proof
  • Corollary 5
  • proof
  • Theorem 6: ngo_new_2002
  • ...and 46 more